The present invention relates to a method of selecting ions in an ion storage device with high resolution in a short time period while suppressing amplitude of ion oscillation immediately after the selection.
In an ion storage device, e.g. a Fourier transformation ion cyclotron resonance system or an ion trap mass spectrometer, ions are selected according to their mass-to-charge (m/e) ratio. While the ions are held within an ion storage space, a special electric field is applied to the ion storage space to selectively eject a part of the ions having specified m/e values. This method, including the storage and selection of ions, is characteristically applied to a type of mass spectrometry called an MS/MS. In an MS/MS mass spectrometry, first, ions with various m/e values are introduced from an ion generator into the ion storage space, and an ion-selecting electric field is applied to the ion storage space to hold within the space only such ions having a particular m/e value while ejecting other ions from the space. Then, another special electric field is applied to the ion storage space to dissociate the selected ions, called precursor ions, into dissociated ions, called fragment ions. After that, by changing the system parameters, the fragment ions created in the ion storage space are ejected toward an ion detector to build a mass spectrum. The spectrum of the fragment ions contains information about the structure of the precursor ions. This information makes it possible to determine the structure of the precursor ions, which cannot be derived from a simple analysis of the m/e ratio. For ions with complex structures, more detailed information about the ion structure can be obtained by a repetition of selection and dissociation of the ions within the ion storage device (MSn analysis).
The special electric field for selecting ions is usually produced by applying voltages having waveforms with opposite polarities to a pair of opposite electrodes which define the ion storage space. The special electric field is produced without changing the ion storage condition. In an ion trap mass spectrometer, voltages having waveforms of opposite polarities are applied to a pair of end cap electrodes, while a radio frequency (RF) voltage is applied to a ring electrode placed between the end cap electrodes. The RF voltage independently determines the ion storage condition.
Each of the ions stored in the ion storage device oscillates at the secular frequency which depends on the m/e value of the ion. When an appropriate electric field for selecting particular ions is applied, the ions oscillate according to the electric field. If the electric field includes a frequency component close to the secular frequency of the ion, the oscillation of the ion resonates to that frequency component of the electric field, and the amplitude gradually increases. After a period of time, the ions collide with the electrodes of the ion storage device or are ejected through an opening of the electrodes to the outside, so that they are evacuated from the ion storage space. In the case of an ion trap mass spectrometer, the secular frequency of an ion in the radial direction differs from that in the axial direction. Usually, the secular frequency in the axial direction is used to remove ions along the axial direction.
Waveforms available for selecting ions include the Stored Waveform Inverse Fourier Transformation (SWIFT; U.S. Pat. No. 4,761,545), Filtered Noise Field (FNF; U.S. Pat. No. 5,134,826), etc. Each of these waveforms is composed of a number of sinusoidal waves with different frequencies superimposed on each other, wherein a frequency component of interest is excluded (this part is called a xe2x80x9cnotchxe2x80x9d). The strength of the ion-selecting electric field produced by the waveform is determined so that ions having such secular frequencies that resonate to the frequency component of the waveform are all ejected from the ion storage space. Ions having secular frequencies equal or close to the notch frequency, which is not contained in the waveform, do not resonate to the electric field. Though these ions might oscillate with a small amplitude, the amplitude does not increase with time, so that the ions are not ejected from the ion storage space. As a result, only such ions that have particular secular frequencies are selectively held in the ion storage space. Thus, the selection of ions is achieved.
However, even if the frequency of the excitation field slightly differs from the secular frequency of the ions, the ions can be excited and the amplitude of the oscillation of the ions increases. This means that the ion selection does not depend solely on whether the waveform contains a frequency component equal to the secular frequency of the ion. Therefore, the notch frequency is determined to have a certain width. However, the ions having a secular frequency at the boundary of the notch frequency are still unstable in oscillation.
As regards the conventional ion-selecting waveforms represented by SWIFT and FNF, past significance has primarily focused on whether the frequency components of the ion-selecting wave include the secular frequency of the ions to be held in the ion storage space.
In a practical mass spectrometry, various processes are performed after the ions are selected. An example of the process is the excitation of precursor ions with an electric field to produce fragment ions, called xe2x80x9cfragmentationxe2x80x9d. In this process, the strength of the excitation field needs to be properly adjusted so as not to eject the precursor ions from the ion storage space. Excessive decrease in the strength of the electric field, however, results in an inefficient fragmentation. Accordingly, the strength of the electric field needs to be controlled precisely. When the initial amplitude of the ion oscillation is large before the excitation field is applied, the ions may be ejected even with a weak electric field. In an ion trap mass spectrometer, the RF voltage needs to be lowered before fragmentation to establish a condition for the fragment ions to be stored. In this process, if the initial amplitude of the oscillation of the precursor ions is large, the motion of the precursor ions becomes unstable, and the ions are ejected from the ion storage space. It is therefore necessary to place a xe2x80x9ccooling processxe2x80x9d for waiting for the oscillation of the precursor ions to subside before fragmentation. Placing such a process consequently leads to a longer time for completing the entire processes, and deteriorates the throughput of the system.
In theory, in an ion trap mass spectrometer, the strength of the RF electric field within the ion storage space determines the secular frequencies of the ions according to their m/e values. In practice, however, the RF electric field deviates slightly from the theoretically designed quadrupole electric field, so that the secular frequency is not a constant value but changes according to the amplitude of the ion oscillation. The deviation of the electric field is particularly observable around a center of the end cap electrodes because they have openings for introducing and ejecting ions. Around the opening, the secular frequency of the ion is lower than that at the center of the ion storage space. In the case of an ion whose secular frequency is slightly higher than the notch frequency, its amplitude increases due to the excitation field when it is at the center of the ion storage space. As the amplitude becomes larger, however, the secular frequency becomes lower, and approaches the notch frequency. This makes the excitation effect on the ion poorer. Ultimately, the amplitude stops increasing at a certain amplitude and begins to decrease.
In the case of an ion whose secular frequency is slightly lower than the notch frequency when it is at the center of the ion storage space, on the other hand, its amplitude increases due to the excited oscillation, and the secular frequency gradually departs from the notch frequency. This increases the efficiency of excitation, and the ion is ultimately ejected from the ion storage space. These cases show that, even if a notch frequency is determined, one cannot tell whether or not ions can be ejected by simply comparing the notch frequency with the secular frequency of the ions, because the interaction is significantly influenced by the strength of excitation field, the dependency of the secular frequency on the amplitude, etc. This leads to a problem that the width of a notch frequency is not allowed to be narrow enough to obtain an adequate resolution of ion selection.
None of the prior art methods presented a detailed theoretical description of the motion of ions in the excitation field: the width of the notch frequency or the value of the excitation voltage has been determined by an empirical or experimental method. To solve the above problem, it is necessary to precisely analyze the motion of ions with respect to time, as well as to think of the frequency components. Therefore, using some theoretical formulae, the behavior of ions in the conventional method is discussed.
First, the equation of the motion of an ion is discussed. In an ion trap mass spectrometer, z-axis is normally determined to coincide with the rotation axis of the system. The motion of an ion in the ion storage space is given by the well-known Mathieu equations. For the convenience of explanation, the motions of ions responding to the RF voltage are represented by their center of RF oscillation averaged over a cycle of RF frequency. The average force acting on the ions is approximately proportional to the distance from the center of the ion storage space (pseudo-potential well model; see, for example, xe2x80x9cPractical Aspects of Ion Trap Mass Spectrometry, Volume 1xe2x80x9d, CRC Press, 1995, page 43). Thus, the equation of motion is given as follows:                                           ⅆ            2                    ⁢          z                          ⅆ                      t            2                              +                        ω          z          2                ⁢        z              =                            f          s                ⁢                  (          t          )                    m                  ω      z        =                  e        ⁢                  xe2x80x83                ⁢        V                              2                ⁢        m        ⁢                  xe2x80x83                ⁢                  z          0          2                ⁢        Ω            
where, m, e and xcfx89z are the mass, charge and secular frequency of the ion, fs(t) is an external force, V and xcexa9 are the amplitude and angular frequency of the RF voltage, and z0 is the distance between the center of the ion trap and the top of the end cap electrode. Similar equations can be applied also to an FITCR system by regarding z as the amplitude from a guiding center along the direction of the excitation of oscillation.
When the external force fs(t) is an excitation field with a single frequency, it is given by                                                         f              s                        ⁡                          (              t              )                                =                                    F              s                        ⁢                          exp              ⁡                              (                                                      jω                    s                                    ⁢                  t                                )                                                                                  =                      e            ⁢                          xe2x80x83                        ⁢                          E              s                        ⁢                          exp              ⁡                              (                                                      jω                    s                                    ⁢                  t                                )                                                          "AutoLeftMatch"
where Fs(=eEs) is the amplitude of the external force, Es is the strength of the electric field produced in the ion storage space by Fs, xcfx89s is the angular frequency of the external force, and j is the imaginary unit. In an actual ion trap mass spectrometer or the like, the strength of the electric field in the ion storage space cannot be thoroughly uniform when voltages of opposite polarities xc2x1vs are applied to the end cap electrodes. In the above equation, however, the strength of the electric field is approximated to be a uniform value Es=vs/z0. The amplitude is represented by a complex number. In a solution obtained by calculation, the real part, for example, gives the real value of the amplitude. Though the arbitrary phase term is omitted in the equation, it makes no significant difference in the result. Similarly, in the following equations, the arbitrary or constant phase term is often omitted.
With the above formula, the equation of motion is rewritten to give the following stationary (particular) solution:                               z          =                      xe2x80x83                    ⁢                                                    F                s                            m                        ⁢                          1                                                ω                  z                  2                                -                                  ω                  s                  2                                                      ⁢                          exp              ⁡                              (                                                      jω                    s                                    ⁢                  t                                )                                                                                  ≅                      xe2x80x83                    ⁢                                                    F                s                                            2                ⁢                m                ⁢                                  xe2x80x83                                ⁢                                  ω                  z                                ⁢                Δω                                      ⁢                          exp              ⁡                              (                                                      jω                    s                                    ⁢                  t                                )                                                          "AutoLeftMatch"
Here, xcex94xcfx89=xcfx89zxe2x88x92xcfx89s is the difference between the frequency of excitation field and the secular frequency of the ion. As for general solution of the equation of motion, the state of motion greatly varies depending on the initial condition of the ion. For example, the condition with initial position z=0 and initial velocity dz/dt=0 brings about an oscillation whose amplitude is twice as large as that of the above stationary solution.
When the secular frequency xcfx89z of an ion is close to the frequency xcfx89s of the excitation field, or when xcex94xcfx89 is small, the oscillation amplitude of the ion increases enough to eject the ion.
As in the case of FNF, when the excitation field is composed of a number of sinusoidal waves superimposed on each other, it is possible to eject all the ions by setting the intervals of the frequencies of the excitation field adequately small, and by giving an adequate strength to the excitation field to eject even such an ion whose secular frequency is located between the frequencies of the excitation field. In order to leave ions with a particular m/e value in the ion storage space, the frequency components close to the secular frequency of the ions should be removed from the excitation field. The motion of the ions, however, is significantly influenced by phases of the frequency components around the notch frequency.
For example, when an ion with a secular frequency of xcfx89z is located at the center of the notch having the width of 2xcex94xcfx89, the frequencies at both sides of the notch are xcfx89zxc2x1xcex94xcfx89. Denoting the phases of the above frequency components by xcfx861 and xcfx862, the waveform composed is represented by the following formula (trigonometric functions are used for facility of understanding):             sin      ⁢              (                                            (                                                ω                  z                                -                Δω                            )                        ⁢            t                    +                      φ            1                          )              +          sin      ⁢              (                                            (                                                ω                  z                                +                Δω                            )                        ⁢            t                    +                      φ            2                          )              =      2    ⁢          sin      ⁢              (                                            ω              z                        ⁢            t                    +                                                    φ                1                            +                              φ                2                                      2                          )              ⁢          cos      ⁢              (                              Δω            ⁢                          xe2x80x83                        ⁢            t                    +                                                    φ                2                            -                              φ                1                                      2                          )            
This formula contains an excitation frequency that is equal to the secular frequency xcfx89z of the ion. Therefore, even when an ion is located at the center of the notch, the ion experiences the excitation. The initial amplitude of the excitation voltage greatly changes according to the envelope of the cosine function depending on the difference 2xcex94xcfx89 between the two frequencies. Thus, the phase of this enveloping function greatly influences the oscillation of the ion. Accurate control of the behavior of the ion is very difficult because of the presence of a greater number of frequency components of the excitation fields outside the notch with their phases correlating to each other.
This suggests that the actual motion of an ion cannot be described based solely on whether a particular frequency is included in the frequency components, or the coefficients of the Fourier transformation, of the excitation waveform. Therefore, when, as in FNF, the excitation field is composed of frequency components with random phases, the correlations of the phases of the frequency components in the vicinity of the notch cannot be properly controlled, so that the selection of ions with high resolution is hard to be performed.
Use of waveforms having harmonically correlated phases, as in SWIFT, may provide one possibility of avoiding the above problem. To allow plural frequency components of the excitation field to act on the ion at a given time point, a complicated control of the phases of the plural frequency components is necessary for harmonization. Therefore, the simplest waveform is obtained by changing the frequency with time. Further, for the convenience of analysis, the changing rate of the frequency should be held constant. Accordingly, the following description about the motion of the ion supposes that the frequency is scanned at a fixed rate.
With xcfx86(t) representing a phase depending on time, let the waveform for selecting ions be given as follows:
fs(t)=Fs exp(jxcfx86(t))
The effective angular frequency xcfx89e(t) acting actually on the ion at the time point t, which is equal to the time-derivative rate of xcfx86(t), is given by                     ω        e            ⁢              (        t        )              ≡                  ⅆ                  φ          ⁢                      (            t            )                                      ⅆ        t              =                              a          ⁢                      xe2x80x83                    ⁢          t                +                  ω          0                    ∴              φ        ⁢                  (          t          )                      =                            a          2                ⁢                  t          2                    +                        ω          0                ⁢        t            +              φ        0            
where xcfx860 and xcfx890 represent the phase and the angular frequency at the time point t=0, respectively, and a represents the changing rate of the angular frequency. The phase xcfx86(t) is thus represented by a quadratic function of time t.
To examine what frequency components are contained in the external force, the formula is next rewritten as follows by the Fourier transformation.                                                                                           xe2x80x83                                ⁢                                                      F                    ⁡                                          (                      ω                      )                                                        =                                                            ∫                                              -                        ∞                                                                    +                        ∞                                                              ⁢                                                                                            f                          s                                                ⁡                                                  (                          t                          )                                                                    ⁢                                              exp                        ⁡                                                  (                                                                                    -                              jω                                                        ⁢                                                          xe2x80x83                                                        ⁢                            t                                                    )                                                                    ⁢                                              ⅆ                        t                                                                                                                                                                    =                                                      F                    s                                    ⁢                                                            ∫                                              -                        ∞                                                                    +                        ∞                                                              ⁢                                                                  exp                        ⁡                                                  (                                                      j                            ⁡                                                          [                                                                                                                                    a                                    2                                                                    ⁢                                                                      t                                    2                                                                                                  -                                                                                                      (                                                                          ω                                      -                                                                              ω                                        0                                                                                                              )                                                                    ⁢                                  t                                                                +                                                                  φ                                  0                                                                                            ]                                                                                )                                                                    ⁢                                              ⅆ                        t                                                                                                                                                                    =                                                      (                                          1                      +                      j                                        )                                    ⁢                                                            π                      a                                                        ⁢                                      F                    s                                    ⁢                                      exp                    ⁡                                          (                                              j                        ⁡                                                  [                                                                                                                    -                                                                  1                                                                      2                                    ⁢                                    a                                                                                                                              ⁢                                                                                                (                                                                      ω                                    -                                                                          ω                                      0                                                                                                        )                                                                2                                                                                      +                                                          φ                              0                                                                                ]                                                                    )                                                                                                          "AutoLeftMatch"                                          xe2x80x83                ⁢                                            f              s                        ⁡                          (              t              )                                =                                    1                              2                ⁢                π                                      ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                F                  ⁡                                      (                    ω                    )                                                  ⁢                                  exp                  ⁡                                      (                                          jω                      ⁢                                              xe2x80x83                                            ⁢                      t                                        )                                                  ⁢                                  ⅆ                  ω                                                                    ⁢                  xe2x80x83                    
This shows that the phase of the Fourier coefficient F(xcfx89) is a quadratic function of the angular frequency xcfx89.
By discretizing the Fourier coefficient F(xcfx89) with the discrete frequencies xcfx89k=kxcex4xcfx89 (k is integer) of interval xcex4xcfx89, fs(t) can be rewritten in the following form similar to SWIFT:                     f        I            ⁢              (        t        )              =                  ∑        k            ⁢                        F          I                ⁢                  exp          ⁢                      (                          j              ⁡                              [                                                                            ω                      k                                        ⁢                    t                                    +                                                            φ                      I                                        ⁢                                          (                      k                      )                                                                      ]                                      )                                                  φ        I            ⁢              (        k        )              =                                        -                          1                              2                ⁢                a                                              ⁢                                    (                                                ω                  k                                -                                  ω                  0                                            )                        2                          +                  φ          0                    =                                    -                          1                              2                ⁢                a                                              ⁢                                    (                                                k                  ⁢                                      xe2x80x83                                    ⁢                  δ                  ⁢                                      xe2x80x83                                    ⁢                  ω                                -                                  ω                  0                                            )                        2                          +                  φ          0                    
This shows that, with discretely defined waveforms for scanning frequencies, the constant phase term xcfx86I(k) of each frequency component is represented as a quadratic function of k. It is supposed here that the two frequency components xcfx89k and xcfx89k+1 take the same value at the time point tk. This condition is expressed as follows:
xcfx89ktk+xcfx86I(k)=xcfx89k+1tk+xcfx86I(k+1)
From this equation, the following equation is deduced:             ω      e        ⁢          (              t        k            )        =                    a        ⁢                  xe2x80x83                ⁢                  t          k                    +              ω        0              =                            ω          k                +                  ω                      k            +            1                              2      
This means that, when two adjacent frequency components are of the same phase and reinforcing each other, the frequency corresponds to the effective frequency of the composed waveform fI(t) at the time point tk. Further, when the interval xcex4xcfx89 is set adequately small, fI(t) becomes a good approximation of the frequency-scanning waveform fs(t). Therefore, the following discussion concerning the continuous waveform fs(t) is completely applicable also to the waveform fI(t) composed of discrete frequency components.
For ease of explanation, the initial condition is supposed as xcfx890=0 and xcfx860=0. This condition still provides a basis for generalized discussion because it can be obtained by the relative shifting of the axis of time to obtain xcfx89s(t)=0 at t=0 and by including the constant phase into Fs. When fs(t) is set not too great, the ions demonstrate a simple harmonic oscillation with an angular frequency of xcfx89z. Accordingly, with the amplitude z represented as a multiplication of a simple harmonic oscillation and an envelope function Z(t) that changes slowly, the equation of motion can be approximated as follows:       z    =                  Z        ⁡                  (          t          )                    ⁢              exp        ⁡                  (                                    jω              z                        ⁢            t                    )                                                                                                                                  ⅆ                    2                                    ⁢                  z                                                  ⅆ                                      t                    2                                                              +                                                ω                  z                  2                                ⁢                z                                      =                          xe2x80x83                        ⁢                                          (                                                                                                    ⅆ                        2                                            ⁢                                              Z                        ⁡                                                  (                          t                          )                                                                                                            ⅆ                                              t                        2                                                                              +                                      2                    ⁢                                          jω                      z                                        ⁢                                                                  ⅆ                                                  Z                          ⁡                                                      (                            t                            )                                                                                                                      ⅆ                        t                                                                                            )                            ⁢                              exp                ⁡                                  (                                                            jω                      z                                        ⁢                    t                                    )                                                                                                      ≅                          xe2x80x83                        ⁢                          2              ⁢                              jω                z                            ⁢                                                ⅆ                                      Z                    ⁡                                          (                      t                      )                                                                                        ⅆ                  t                                            ⁢                              exp                ⁡                                  (                                                            jω                      z                                        ⁢                    t                                    )                                                                          "AutoLeftMatch"  
The term of the external force is given as follows:                     f        s            ⁢              (        t        )              m    =                    F        s            m        ⁢          exp      ⁢              (                  j          ⁢                      a            2                    ⁢                      t            2                          )            
With this formula, the equation of motion can be further rewritten as follows:             ⅆ              Z        ⁢                  (          t          )                            ⅆ      t        =                    F        s                    2        ⁢        j        ⁢                  xe2x80x83                ⁢        m        ⁢                  xe2x80x83                ⁢                  ω          z                      ⁢          exp      ⁢              (                  j          ⁡                      [                                                            a                  2                                ⁢                                  t                  2                                            -                                                ω                  z                                ⁢                t                                      ]                          )            
Supposing that the coefficient Fs of the external force takes a constant value F0 irrespective of time, and that the initial amplitude Z(xe2x88x92∞)=0, the envelope function is obtained as follows:                                           Z            ⁡                          (              t              )                                =                                                    F                0                                            2                ⁢                j                ⁢                                  xe2x80x83                                ⁢                m                ⁢                                  xe2x80x83                                ⁢                                  ω                  z                                                      ⁢                                          ∫                                  -                  ∞                                t                            ⁢                                                exp                  ⁡                                      (                                          j                      ⁡                                              [                                                                                                            a                              2                                                        ⁢                                                          τ                              2                                                                                -                                                                                    ω                              z                                                        ⁢                            τ                                                                          ]                                                              )                                                  ⁢                                  ⅆ                  τ                                                                                                  =                                                    F                0                                            2                ⁢                j                ⁢                                  xe2x80x83                                ⁢                m                ⁢                                  xe2x80x83                                ⁢                                  ω                  z                                                      ⁢                                          π                a                                      ⁢                                          exp                ⁡                                  (                                                            -                      j                                        ⁢                                                                  ω                        z                        2                                                                    2                        ⁢                        a                                                                              )                                            ⁡                              [                                                      C                    ⁡                                          (                      u                      )                                                        +                                      j                    ⁢                                          xe2x80x83                                        ⁢                                          S                      ⁡                                              (                        u                        )                                                                              +                                                            1                      2                                        ⁢                                          (                                              1                        +                        j                                            )                                                                      ]                                                          ⁢      "AutoLeftMatch"          
        ⁢          u      =                                                  a              ⁢                              xe2x80x83                            ⁢              t                        -                          ω              z                                                          a              ⁢                              xe2x80x83                            ⁢              π                                      =                                                            ω                e                            ⁡                              (                t                )                                      -                          ω              z                                                          a              ⁢                              xe2x80x83                            ⁢              π                                          
where C(u) and S(u) are the Fresnel integrals, and the term in the square brackets represents the length of the line connecting the points (xe2x88x92xc2xd, xe2x88x92xc2xd) and (C(u), S(u)) on the complex plane as shown in FIG. 2.
When the effective angular frequency xcfx89e(t) is equal to the secular frequency xcfx89z of the ion, the parameter is u=0, which represents the origin in FIG. 2. Application of the frequency-scanning waveform moves the point (C(u), S(u)) to (+xc2xd, +xc2xd), where the term in the square brackets is (1+j) and the residual amplitude Z(+∞) of the ion oscillation is given as follows:       Z    ⁢          (              +        ∞            )        =                              F          0                          2          ⁢          j          ⁢                      xe2x80x83                    ⁢          m          ⁢                      xe2x80x83                    ⁢                      ω            z                              ⁢              (                  1          +          j                )            ⁢                        π          a                    ⁢              exp        ⁡                  (                                    -              j                        ⁢                                          ω                z                2                                            2                ⁢                a                                              )                      ≡          Z      max      
This calculation corresponds to the case where the excitation field is applied without any notch, because the amplitude coefficient of the excitation waveform is given the constant value F0. The residual amplitude Z(+∞)=Zmax is almost constant irrespective of the mass m because m and xcfx89z are almost inversely proportional to each other. When F0 is determined so that the absolute value of the envelope function |Zmax| becomes greater than the size z0 of the ion storage space, any ion with any m/e value is ejected from the ion storage space. In an ion trap mass spectrometer, the actual oscillation of ions takes places around the central position defined by the pseudo-potential well model, with the amplitude of about (qz/2)z and the RF frequency of xcexa9, where qz is a parameter representing the ion storage condition, written as follows:       q    z    =            2      ⁢      e      ⁢              xe2x80x83            ⁢      V              m      ⁢              xe2x80x83            ⁢              z        0        2            ⁢              Ω        2            
This shows that the maximum amplitude is about |Z(+∞)|(1+qz/2). It should be noted that this amplitude becomes larger as the mass number of the ion is smaller and qz is accordingly greater.
When the waveform for exciting ions has a notch, the amplitude coefficient Fs is described as a function of time t or a function of effective frequency xcfx89e(t)=at. The conventional techniques, however, employ such a simple method that the amplitude of the frequency components inside the notch is set at zero. That is, Fs is given as follows (FIG. 3):             F      s        ⁡          (      t      )        =      {                                                                      xe2x80x83                            ⁢                              F                0                                                          ⋯                                                              xe2x80x83                            ⁢                                                t                  ≤                                      t                    1                                                  ,                                                      t                    2                                    ≤                  t                                                                                                                        xe2x80x83                            ⁢              0                                            ⋯                                                              xe2x80x83                            ⁢                                                t                  1                                 less than                 t                 less than                                   t                  2                                                                        "AutoLeftMatch"      
Since no external force exists in the time period t1 less than t less than t2, the envelop function after the application of the excitation waveform, i.e. the residual amplitude Z(+∞), is represented by a formula similar to the aforementioned one, as shown below:                                           Z            ⁡                          (                              +                ∞                            )                                =                      xe2x80x83                    ⁢                                                    F                0                                            2                ⁢                j                ⁢                                  xe2x80x83                                ⁢                m                ⁢                                  xe2x80x83                                ⁢                                  ω                  z                                                      ⁢                                          ∫                                  -                  ∞                                                  t                  1                                            ⁢                              +                                                      ∫                                          t                      2                                                              +                      ∞                                                        ⁢                                                            exp                      ⁡                                              (                                                  j                          ⁡                                                      [                                                                                                                            a                                  2                                                                ⁢                                                                  τ                                  2                                                                                            -                                                                                                ω                                  z                                                                ⁢                                τ                                                                                      ]                                                                          )                                                              ⁢                                          ⅆ                      τ                                                                                                                                        =                      xe2x80x83                    ⁢                                                    F                0                                            2                ⁢                j                ⁢                                  xe2x80x83                                ⁢                m                ⁢                                  xe2x80x83                                ⁢                                  ω                  z                                                      ⁢                                          π                a                                      ⁢                          exp              ⁡                              (                                                      -                    j                                    ⁢                                                            ω                      z                      2                                                              2                      ⁢                      a                                                                      )                                      xc3x97                          [                                                (                                      1                    +                    j                                    )                                -                                                                                              xe2x80x83                    ⁢                                    {                                                C                  ⁡                                      (                                          u                      2                                        )                                                  +                                  j                  ⁢                                      xe2x80x83                                    ⁢                                      S                    ⁡                                          (                                              u                        2                                            )                                                                      -                                  C                  ⁡                                      (                                          u                      1                                        )                                                  -                                  j                  ⁢                                      xe2x80x83                                    ⁢                                      S                    ⁡                                          (                                              u                        1                                            )                                                                                  }                        ]                                "AutoLeftMatch"
where u1 and u2 are the parameters of the Fresnel functions at time points t1 and t2. Similar to the case of the excitation waveform with no notch, the term in the last square brackets represents the vector sum of the two vectors: one extending from (xe2x88x92xc2xd, xe2x88x92xc2xd) to (C(u1), S(u1)) and the other extending from (C(u2), S(u2)) to (xe2x88x92xc2xd, xe2x88x92xc2xd) in FIG. 2. In other words, the value represents the vector subtraction where the vector extending from (C(u1), S(u1)) to (C(u2), S(u2)) is subtracted from the vector extending from (xe2x88x92xc2xd, xe2x88x92xc2xd) to (xe2x88x92xc2xd, xe2x88x92xc2xd). When u1 and u2 are located in opposition to each other across the origin, or when u2=xe2x88x92u1 greater than 0, the residual amplitude |Z(+∞)| is smaller than Zmax of the no-notch case. As the value of u2 (=xe2x88x92u1) increases, the value of |Z(+∞)| decreases. The rate of decrease, however, is smaller when u2 (=xe2x88x92u1) is greater than 1.
For the selection of ions, t1 and t2 are determined so that the secular frequency xcfx89Z of the target ions to be left in the ion storage space comes just at the center of the frequency range of the notch: xcfx89e(t1) to xcfx89e(t2). That is, the frequency xcfx89cxe2x89xa1xcfx89e(tc)=(xcfx89e(t1)+xcfx89e(t2))/2 at the time point tcxe2x89xa1(t1+t2)/2 is made equal to xcfx89z. Under this condition, the residual amplitude |Z(+∞)| is so small that it does not exceed the size of the ion storage space, so that the ions are kept stored in the ion storage space. Increase in the width of the notch, or in the distance between xcfx89e(t1) and xcfx89e(t2), provides a broader mass range for the ions to remain in the ion storage space and hence deteriorates the resolution of ion selection. Therefore, the width of the notch should be set as narrow as possible. The narrower notch, however, makes the residual amplitude |Z(+∞)| larger, which becomes closer to the value of the no-notch case. When the width of the notch is further decreased, the ions to be held in the ion storage space are ejected from the space together with other ions to be ejected. Accordingly, to obtain a high resolution of ion selection, the scanning speed a of the angular frequency needs to be set lower to make {square root over (axcfx80)} smaller, in order to make |u| greater, while maintaining the frequency difference |xcfx89e(t)xe2x88x92xcfx89z| small. This requires a longer time period for scanning the frequency range, from which arises a problem that the throughput of the system decreases due to the longer time period for performing a series of processes.
When u1=xe2x88x921 and u2=+1, the value of the term in the square brackets (i.e. length) is about 0.57, which cannot be regarded as small enough compared to 1.41 which is the absolute value of the term in the square brackets for the ions outside the notch. For example, unnecessary ions outside the notch are ejected from the ion storage space when the excitation voltage is adjusted so that the residual amplitude Zmax after the application of the selecting waveform is 1.41z0. In this case, the ion to be held in the space, having its secular frequency equal to the frequency xcfx89c at the center of the notch, has the residual amplitude of 0.57z0. Though the ion is held in the ion storage space, its motion is relatively unstable. The maximum amplitude increases to about 0.75z0 during the application of the selecting waveform, reaching the region where the secular frequency of the ion changes due to the influence of the hole of the end cap electrode. Thus, under a certain initial condition, the ion is ejected from the ion storage space.
When u1=xe2x88x920.5 and u2=+0.5, the scanning speed of the angular frequency is increased fourfold, and the time required for scanning the frequency is shortened to a quarter. In this case, the ion to be held in the space, having its secular frequency equal to the frequency xcfx89c at the center of the notch, has a residual amplitude of 0.87z0, and almost all the ions are ejected during the application of the selecting waveform.
As explained above, the conventional methods are accompanied by a problem that the resolution of ion selection cannot be adequately improved within a practical time period of ion selection. In other words, an improvement in the resolution of ion selection causes an extension of the time period of ion selection in proportion to the second power of the resolution.
Another problem is that the ions, oscillating with large amplitude immediately after the application of the ion-selecting waveform, are very unstable because they are dissociated by the collision with the molecules of the gas in the ion storage space. Also, an adequate cooling time is additionally required for damping the oscillation of the ions before the start of the next process.
Still another problem is that, when the excitation field is composed of frequency components with random phases, as in the FNF, the phases of the frequency components in the vicinity of the notch cannot be properly controlled, so that it is difficult to select ions with high resolution.
The present invention addresses the above problems, and proposes a method of selecting ions in an ion storage device with high resolutions in a short time period while suppressing oscillations of ions immediately after the selection.
To solve the above problems, the present invention proposes a method of selecting ions in an ion storage device with high resolution in a short period of time while suppressing amplitude of ion oscillation immediately after the selection. In a method of selecting ions within a specific range of mass-to-charge ration by applying an ion-selecting electric field in an ion storage space of an ion storage device, the ion-selecting electric field is produced from a waveform whose frequency is substantially scanned within a preset range, and the waveform is made anti-symmetric at around a secular frequency of the ions to be left in the ion storage space.
One method of making the waveform anti-symmetric is that a weight function, whose polarity reverses at around the secular frequency of the ions to be left in the ion storage space, is multiplied to the waveform.
Another method of making the waveform anti-symmetric is that a value of (2k+1)xcfx80(k is an arbitrary integer) is added to the phases of the waveforms.
It is preferable that the frequency scanning of the waveform is performed in the direction of decreasing the frequency. Further, series of waveforms with different scanning speeds may be used to shorten the time required for the selection.
The residual amplitude of the ions that are left in the ion storage space after the ion-selecting waveform is applied can be suppressed by slowly changing the weight function of the amplitude at the boundary of the preset frequency range to be scanned. The form of the notch can be designed arbitrarily as long as the weight function is anti-symmetric across the notch frequency.
FIG. 1 shows an example of the ion-selecting waveform fs(t) according to the present invention and the weight function Fs(t) for producing the above waveform.
The waveform according to the present invention is characteristic also in that the ion selection can be performed even with a zero width of the notch frequency.
The above-described ion-selecting waveforms whose frequency is substantially scanned is composed of plural sinusoidal waves with discrete frequencies, and each frequency component of the waveform has a constant part in its phase term which is written by a quadratic function of its frequency or by a quadratic function of a parameter that is linearly related to its frequency.